Absence of critical scaling in the Schelling segregation model
Abstract
We find no evidence of critical scaling in the Schelling segregation model, in either the Moore neighborhood or its dense-spectrum extension to Chebyshev radii up to r0 = 6 (k = 168 neighbors). On periodic grids up to L = 320 with 50 trials per point (> 12,500 runs), every finite-size scaling diagnostic in the Moore baseline fails: the per-L Tc does not drift, Var(S) ∼ L−2.02 ± 0.09 matches trivial averaging, γ/ν≈ 0, and the scaling collapse never reaches a finite optimum. The 8-site Moore neighborhood restricts satisfaction to ratios j/k with k ≤ 8, giving S(T) a staircase structure with 23 rational thresholds; discreteness alone does not forbid criticality (cf. the Ising model), but the scaling evidence rules it out empirically. A branching-ratio calculation predicts subcritical cascades of mean size 1/(1−R) and is validated by perturbation experiments to within 15%; the multiscalar dissimilarity length stays finite across the transition. The dense-spectrum extension strengthens the negative verdict: across r0 ∈ 3,4,5,6 on L ∈ 40,80,160 the Binder cumulant has no L-curve crossing and the per-L Tc drift is monotonic and unsaturated; at r0 = 4, extending to L = 320 gives α= −2.70, below the critical boundary α= −2, dissolving an apparent α= +0.81 signal visible only on L ∈ 40,80. The mechanism is the absence of long-range correlation in equilibrium plus deterministic high-k dynamics, not the staircase structure. With a Beta-distributed heterogeneous tolerance, the intolerant tail drives segregation even at moderate population-average tolerance. The staircase theorem and cascade mechanism together account for the Schelling transition without invoking critical phenomena.
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BibTeX
@misc{rifaki2026absence,
title = {{Absence of critical scaling in the Schelling segregation model}},
author = {Mouhssine Rifaki},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.16557},
eprint = {2608.16557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16557},
}
RIS
TY - UNPB
TI - Absence of critical scaling in the Schelling segregation model
AU - Mouhssine Rifaki
PY - 2026
DA - 2026/08/17
PB - arXiv
DO - 10.48550/arxiv.2608.16557
UR - https://arxiv.org/abs/2608.16557
AB - We find no evidence of critical scaling in the Schelling segregation model, in either the Moore neighborhood or its dense-spectrum extension to Chebyshev radii up to r_0 = 6 (k = 168 neighbors). On periodic grids up to L = 320 with 50 trials per point (> 12,500 runs), every finite-size scaling diagnostic in the Moore baseline fails: the per-L T_c does not drift, Var(S) ∼ L^−2.02 ± 0.09 matches trivial averaging, γ/ν≈ 0, and the scaling collapse never reaches a finite optimum. The 8-site Moore neighborhood restricts satisfaction to ratios j/k with k ≤ 8, giving S(T) a staircase structure with 23 rational thresholds; discreteness alone does not forbid criticality (cf. the Ising model), but the scaling evidence rules it out empirically. A branching-ratio calculation predicts subcritical cascades of mean size 1/(1−R) and is validated by perturbation experiments to within 15%; the multiscalar dissimilarity length stays finite across the transition. The dense-spectrum extension strengthens the negative verdict: across r_0 ∈ 3,4,5,6 on L ∈ 40,80,160 the Binder cumulant has no L-curve crossing and the per-L T_c drift is monotonic and unsaturated; at r_0 = 4, extending to L = 320 gives α= −2.70, below the critical boundary α= −2, dissolving an apparent α= +0.81 signal visible only on L ∈ 40,80. The mechanism is the absence of long-range correlation in equilibrium plus deterministic high-k dynamics, not the staircase structure. With a Beta-distributed heterogeneous tolerance, the intolerant tail drives segregation even at moderate population-average tolerance. The staircase theorem and cascade mechanism together account for the Schelling transition without invoking critical phenomena.
ER -